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Maths ppt_100731.pptx

This document discusses methods for solving algebraic and transcendental equations. It describes directive or direct methods that can find exact solutions, such as factorization. It also describes iterative or indirect methods that can find approximate solutions, including the bisection method and Regula Falsi method. The bisection method repeatedly bisects an interval to narrow in on a solution, while the Regula Falsi method uses linear interpolation to refine the interval containing the root. Examples of each method's algorithm are provided along with graphical representations.

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Madhav Institute OF Technology & Science, Gwalior (A Govt. Aided UGC Autonomous & NAAC Accredited Institute Affiliated to RGPV, Bhopal) Information Technology Department Proficiency Test Engineering MATHEMATICS - III (100028) Semester-4 Submitted By- Niranjan Batham (0901IT211036) Parth Joshi (0901IT211037) Prabhat Uikey (0901IT211039) Prakhar Gupta (0901IT211040) Prashant Mourya (0901IT211041) Priyank Singh (0901IT211042) Submitted To- Dr. J K Muthele
SOLUTION OF ALGEBRAIC AND TRANSCENDENTAL EQUATIONS • Consider the equation of the form f(x)=0. • If f(x) is a quadratic, cubic or biquadratic expression then algebraic formulae are available for expressing the roots. • But when f(x) is a polynomial of higher degree or an expression involving transcendental functions e.g., 1 + cos x – 5x, x tan x – cosh x, e-x – sin x etc., algebraic methods are not available.
In order to solve these type of equations following methods exist: • DIRECTIVE METHODS: The methods which are used to find solutions of given equations in the direct process is called as directive methods. Example: remainder theorem, Factorization method etc Note: By using Directive Methods, it is possible to find exact solutions of the given equation. • ITERATIVE METHODS (INDIRECT METHODS): The methods which are used to find solutions of the given equation in some indirect process is called as Iterative Methods Note: By using Iterative methods, it is possible to find approximate solution of the given equation and also it is possible to find single solution of the given equation at the same time. Methods
The bisection method in mathematics is a root finding method which repeatedly bisects an interval and then selects a subinterval in which a root must lie for further processing. It is a very simple and robust method, but it is also relatively slow. Because of this, it is often used to obtain a rough approximation to a solution which is then used as a starting point for more rapidly converging methods . Bisection Method Step 1: Choose two approximations A and B (B>A) such that f(A)*f(B)<0 Step 2: Evaluate the midpoint C of [A,B] given by C=(A+B)/2 • Step 3: If f(C)*f(B)<0 then rename B & C as A & B. If not rename of C as B . Then apply the formula of Step 2. • Step 4:Stop evolution when the different of two successive values of C obtained from Step 2 is numerically less than E, the prescribed accuracy Algorithm
Graphical Representation of Bisection Method
REGULA FALSI METHOD • This is the oldest method for finding the real roots of a numerical equation. • It is sometimes known as method of linear interpolation.  Locate the interval (a, b) in which the root lies.  First Approximation to the root  Locate the next interval (a, X0) or ( X0, b ) in which root lies.  Repeat the process until the root converges.
Graphical Representation of Regula Falsi Method
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